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ABSTRACT
A PHARMACOKINETIC MODEL OF THE LIVER
by Samirbhai Rameshbhai Patel
The liver is the major organ where chemical breakdown of organic and inorganic
compounds takes place. A pharmacokinetic heterogeneous model was developed for the
local anesthetic and antiarrhythmic drug lidocaine. It was assumed that transport and
elimination of lidocaine and its metabolites are linear with concentration. Simulations
were done using VisSim software on a 486 based personal computer to obtain
concentration profiles for hepatic clearance and for step input of the drug and its
metabolites. It was found that the rate of uptake and rate of breakdown were 0.48
min and 0.49 m respectively for lidocaine. The results are consistent with
published data. Similar simulations were done for the lidocaine metabolites MEGX and
3-OHLID.
The same heterogeneous model was used with appropriate changes to include
cellular inactivation of the drug and urine output in order to model another
antiarrthymic drug, procainamide. Simulations were done for hepatic clearance and for
bolus (impulse) input to the model. The rate of uptake and rate of release for
procainamide were found to be 9.35 min-1. The rate of breakdown and urine output
were 0.0053 min-1 and 0.0013 min-1 respectively.
Finally, a parameter sensitivity study was done for both lidocaine and its
metabolites and procainamide in order to determine sensitivity of the model to the
parameters used. The maximum values of rate constants for which the model can
operate in stable range were determined.
A PHARMACOKINETIC MODEL OF THE LIVER
by Samirbhai Rameshbhai Patel
A Thesis Submitted to the Faculty of
New Jersey Institute of Technology in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Biomedical Engineering
Biomedical Engineering Committee
January 1995
APPROVAL PAGE
A PHARMACOKINETIC MODEL OF THE LIVER
Samirbhai Rameshbhai Patel
Dr. Arthur Ritter, Thesis Advisor Associate Professor of Physiology UMD-New Jersey Medical School
Dr. David Kristol, Committee Member Professor of Chemistry and Director of Biomedical Engineering Program, NJTT
Dr. Stanley S. Reisman, Committee Member Professor of Electrical Engineering and Associate Chairperson for Graduate Studies, NJIT
BIOGRAPHICAL SKETCH
Author: Samirbhai Rameshbhai Patel
Degree: Master of Science in Biomedical Engineering
Date: January 1995
Date of Birth:
Place of Birth:
Undergraduate and Graduate Education:
■ Master of Science in Biomedical Engineering New Jersey Institute of Technology, Newark, New Jersey, 1995
■ Bachelor of Science in Electronics and Communication Engineering University of Madras, Madras, India, 1992
Major: Biomedical Engineering
This thesis is dedicated to my grandparents Maganbhai and Shantaben Patel
v
ACKNOWLEDGMENT
The author wishes to express his sincere gratitude to his advisor, Dr. Arthur Ritter,
for his guidance, friendship, and moral support throughout this research.
The author is grateful and appreciates the timely help and suggestion by Dr. David
Kristol. His advice and help were the basis of my education at NJIT.
Special thanks to Dr. Stanley S. Reisman for serving as a member of the
committee.
vi
TABLE OF CONTENTS
Chapter Page
1 INTRODUCTION 1
1.1 Liver Physiology and Functions 1
1.1.1 Physiology of Liver
1.1.2 Functions of Liver 4
2 METHODS 5
2.1 Types of Models Available 5
2.1.1 Non-Parametric Models 6
2.1.2 Homogenous Models 6
2.1.3 Heteregenous Models 7
2.2 Model for Lidocaine Metabolism 8
2.2.1 Assumptions and Equations 8
2.2.2 Calculations 11
2.2.3 Data Analysis 13
2.3 Model for Procainamide 15
2.3.1 Calculations 16
3 RESULTS AND DISCUSSIONS 18
3.1 Lidocaine and its Metabolities 18
3.1.1 Washout Simulations 18
3.1.2 Sensitivity Study 19
3.1.2(a) Sensitivity Study of Lidocaine 20
3.1.2(b) Sensitivity Study of MEGX and 3-OHLID 20
3.2 Procainamide Simulations 23
3.2.1 Washout Profile 23
vii
TABLE OF CONTENTS (Continued)
Chapter Page
3.2.2 Simulations with Input 23
3.2.3 Sensitivity Study for k12 24
3.2.4 Sensitivity Study for k21 25
3.2.5 Sensitivity Study for 'u' 26
3.2.6 Sensitivity Study for kr 26
4 CONCLUSIONS AND SUGGESTIONS
4.1 For Lidocaine and its Metabolities 28
4.2 For Procainamide 28
4.3 Future Work 29
APPENDIX A Simulation Results 10
APPENDIX B VisSim Description 55
REFERENCES 59
viii
LIST OF TABLES
Table Page 2.1 Comparison of kinetic constants values given in the reference
and predicted after exponential peeling method 13
2.9 Predicted rate constant values for lidocaine metabolities 14
3.1 The range of values for k17, 1(91, kr for lidocaine and its metabolities 21
3.2 Maximum values of both k12 and k91 which can be used together 21
3.3 Normal values of kinetic constants for lidocaine and its metabolities 22
3.4 Procainamide simulations results with different ranges 27
ix
LIST OF FIGURES
Figure Page
1.1 Schematic diagram of liver structure 2
2.1 Compartmental model for hepatic elimination of lidocaine and its metabolities . . . 9
2.2 Metabolities formation of lidocaine 10
2.3 Metabolities formation of procainamide 15
2.4 Compartmental model for hepatic elimination of procainamide 16
x
CHAPTER 1
INTRODUCTION
The liver is the primary organ for metabolism of a wide range of lipophilic compounds,
including many drugs, aromatic hydrocarbons, and chlorinated compounds [1]. Within
the liver, a number of different physiological processes combine to give observed
differences in drug elimination. Net changes in concentration of these compounds are due
to a series of steps [2] summarized as follows;
(1) The net flow of blood (perfusion medium) in the vascular network of the organ.
(2) Uptake of the substrate from the vasculature into the liver cells.
(3) Enzyme mediated reactions within the cells.
(4) Release of metabolic products and unconverted substrate from the cells into the
sinusoids.
It is necessary to understand each of these processes to fully comprehend the
overall process of drug elimination. Thus, a drug elimination model was developed by
considering the above processes either individually or by grouping some of them with
appropriate approximations. The major purpose of this model is to mathematically
describe the dominant physiological processes that are occurring. The distribution of
most drugs from the blood into highly perfused tissues such as liver and kidney is known
to be quite rapid and that distribution in to moderately to poorly perfused tissues such as
muscles and fat is rather slow [3]. This study is part of the whole body pharmacokinetic
model for blood flow. Anesthetic and antiarrthymic drug model of liver was developed.
1.1 Liver Physiology and Functions
The liver is located in the blood stream between the gastrointestinal tract and spleen and
vena cava leading to the heart. The liver can be regarded as a three dimensional, slowly
2
moving lake of blood in which sheets of liver cells, or hepatocytes float [4]. Blood
enters the organ through both the portal vein and the hepatic artery and leaves the liver
via the hepatic vein (Figure 1.1). The portal vein, hepatic vein, and hepatic artery are all
subject to branching. Individual branches serve each lobe of the liver [5]. The blood flow
to the liver is normally about 25% of the cardiac output. The flow is derived from the
two sources, the portal vein and the hepatic artery. The portal vein provides about three
fourth of the blood flow, after passing through, the gastrointestinal capillary bed. The
hepatic artery delivers the remaining one fourth of the blood [6].
Figure 1.1 Schematic diagram of the Liver structure (From B. A. Saville, M. R. Gray, Y. K. Tam, "Models of Hepatic drug elimination", Drug Metabolism Reviews, 24(1), 49- 88, 1992)
1.1.1 Physiology of Liver
The basic functional unit of the liver is the liver lobule, which is a cylindrical structure.
The human liver contains 50,000 to 100,000 individual lobules. The venous sinusoids are
lined by two types of cells (1) Typical endothelial cells and (2) large kupffer cells, which
are tissue microphages. The endothelial lining of the venous sinusoids has extremely
large pores. Beneath this lining, lying between the endothelial cells and hepatic cells, is a
very narrow tissue space called the space of Disse. Because of the large pores in the
endothelium, substances in the plasma move freely into the space of Disse. Kupffer cells
are capable of phagocytizing bacteria and other foreign matter in the blood [7].
Orally administered substances are absorbed in the gastrointestinal tract, and enter
the liver through the portal vein. Drugs administered intravenously circulate through the
body and enter the liver through both hepatic artery and portal vein [2]. The liver
contains about 15% of total blood volume of the body. The liver constitutes an important
blood reservoir in humans.
The liver is a continuous meshwork of highly interconnected blood spaces. The
sinusoidal channels are of varying length and are oriented in a nearly random fashion.
Mixing can occur at various sites within the liver, especially at each node corresponding
to the divergence and convergence of a number of sinusoids. Furthermore, back mixing
within the organ occurs due to flow reversal in the sinusoids. Hepatic metabolism is
highly dependent upon the physical properties of the substrate relative to the physical
characteristics and intrinsic activity of the liver. Every hepatocyte is exposed to the
blood, since the plates of hepatocytes are no more than two cells thick. The hepatocytes
remove substances from the blood and secrete them in to bilary canaliculi lying between
the adjacent hepatocytes. The countercurrent relationship between the bile flow and
blood flow within the lobule minimizes the concentration to the liver's efficiency in
extracting substances from the blood [8]. To investigate the vast range of phenomena that
affect hepatic metabolism, several classes of compounds can be used as probes of
intrahepatic mixing, interphase transport and elimination [2]. The intimate contact of a
large fraction of the hepatocyte surface with blood contributes to the ability of the liver
to clear the blood effectively of certain classes of compounds. The organ must be
considered as a heterogeneous system to account for the transport between the
vasculature and the cells W.
1.1.2 Functions of Liver
The basic functions of the liver can be divided into (1) vascular functions for storage and
filtration of blood (2) metabolic functions concerned with the majority of the metabolic
systems of the body, and (3) secretory and excretory functions that are responsible for
forming the bile that flows through the bile ducts into the gastrointestinal tract. The liver
performs a wide variety of functions. Some of them are endocrine functions, clotting
functions, digestive functions, organic and cholesterol metabolism, excretory and
degradative functions [9]. The liver regulates the metabolism of carbohydrates, lipids and
proteins. The liver transforms and excretes a large number of drugs and toxins. Drugs
and toxins are frequently converted to inactive forms by reactions that occur in
hepatocytes. Drug transformations that occur in hepatocytes are oxidative, acetlyation,
methylation, reduction, and hydrolysis. The transformation that occurs in the liver
renders many drugs more water soluble, and thus they are more readily excreted by the
kidneys. Some drug metabolites are secreted into bile. The liver has a very active
chemical medium and this medium is well known for capability of detoxifying or
excreting into the bile many different drugs including sulfonamides, penicillin,
ampicillin, erythromycin, and many others. In a similar manner several of the different
hormones secreted by the endocrine glands are either chemically altered or excreted by
the liver, including thyroxin.
A heterogeneous model was used to study hepatic elimination of the liphophilic
compound lidocaine and its metabolites. Simulations were done on a 486 based personal
computer with the VisSim (Visual Solution Inc., Westford, MA) graphical modeling
program. A sensitivity study was done which established ranges of kinetic constant
values that can be used. The heterogeneous model had to be changed slightly to simulate
the antiarrthymic drug procainamide. Simulations were done for procainamide and range
of kinetic constant values were determined.
CHAPTER 2
METHODS
The following chapter documents the methods available for modeling kinetics of liver
metabolism. The method used is described in detail. Equations and parameter values are
described. Simulations were done on VisSim software [10].
2.1 Types of Models Available
There are three types of models available for liver metabolism; which include
(1) Non-parametric models,
(2) Homogenous mixing models,
(3) Heterogeneous models
Hepatic metabolism models may be classified according to the assumption made in the
liver physiology and degree of mixing within the organ. Non-parametric models are the
simplest kind of models. These models do not have any adjustable parameter to describe
extent of mixing and interphase transport. A second class of models are homogenous and
these models take into account the extent of intra hepatic mixing. However these models
do not explicitly account for mass transfer between the cells and the vasculature. Both
non-parametric models and homogenous models are described by first order ordinary
differential equations. The heterogeneous models are divided into two types. The first
type uses homogenous compartments to describe intrahepatic mixing and this type
assumes complete mixing on a microscopic level. The second type assumes little or no
mixing on a microscopic level [2]. Microscopic heterogeneous models (no mixing) use
partial differential equations and are distributed parameter models. These models are
more complex then the earlier types described.
5
2.1.1 Non-Parametric Models
The most commonly used and most extensively investigated models of hepatic
elimination are the well stirred model and parallel tube model. Both models ignore the
heterogeneous nature of the liver, assuming that compounds are equally and
instantaneously distributed between the vascular and cellular regions of the liver. Both
models can be mathematically described by the same Taylor series expansion [11]. A
well mixed model is mathematically simple. However, it cannot describe the metabolism
of materials which are limited by the transport into the hepatocytes. A parameter model
is more complex mathematically. Pang and Rowland [12] demonstrated that weak
dependence of the lidocaine metabolism upon flow rate was best described by the well
mixed model. These types of models are mathematically simple and cannot be used to
describe intra hepatic mixing. Well mixed and parallel tube model, require so many
approximations as to limit their use to a small selected number of substrates. Non
parametric models can be described by ordinary differential equations.
2.1.2 Homogenous Models
These types of models are mixing models, and are used to describe mixing phenomena
that occur in the liver. There are two single parameter homogenous models for hepatic
mixing (1) The axial dispersion model [11, 13-16]
(2) The series compartment model [17] •
Both these models are based upon well established models for chemical reactors [18].
The axial dispersion model assumes that the liver is a packed bed in which different
degrees of axial mixing can occur. The dispersion model is mathematically complex and
is used mainly as an empirical model. The series compartment model is mathematically
simple and is very general for species without transport limitations. Axial dispersion
models are described by partial differential equations. The series compartment model
7
does not include the effect of transport processes between the vascular and cellular
regions of the liver. The disadvantage with both types of homogenous models is that
there is no direct means of accounting for physiological phenomena such as interphase
mass transport and variable flow through the liver.
2.1.3 Heterogeneous Model
The heterogeneous model is required to properly investigate transport process in the liver
[2]. Heterogeneous models are typically more complex than homogenous models.
Several models for heterogeneous metabolism have been developed with varying degree
of complexity. Heterogeneous models explicitly account for transport processes between
species in the cells and the vasculature. The heterogeneous models proposed by Tsuji
[19] were used for a variety of antibiotics but this model has drug specific assumptions
and cannot be used as a general model. The Deans-Levich model [20] and the finite stage
transport model [21] are promising but require modifications to be made before they can
be used for liver metabolism. The compartment model proposed by Weisiger [22] has
several advantages such as mathematical simplicity, requires minimum data and is easy
to use as a predictive model for both drugs and metabolites. This model can account for
several physiological processes such as cellular uptake and release as well as intracellular
reactions. The compartment model proposed by Weisiger [22] and modified by Saville
[1] comes closest to the ideal model for liver metabolism. The modified model can be
used to describe time dependent profiles of a variety of compounds. Experimental
evidence suggests that this is particularly effective for species subject to passive
transport. Hence, this model was used for the liver metabolism. The model is
mathematically simple, and can account for a wide variety of physiological phenomena
and is easy to use both for correlating data and as a predictive model [2].
8
The heterogeneous model proposed by Weisiger [22] and modified by Saville [1]
produced a best fit of the data and was used to model hepatic elimination of lidocaine and
its metabolism. A similar model was used to model hepatic elimination of procainamide.
There are two types of enzyme inactivation. (1) Chemical moves in to the cell and
enzymes breakdown chemical in the cell. (2) Enzymes moves outsides cell and
breakdown of chemical occurs outside cell. In lidocaine breakdown occurs inside cell.
While in procainamide breakdown occurs outside cell.
2.2 Model for Lidocaine Metabolism
The liver model for the lipophillic compound lidocaine consists of two compartments: an
extracellular compartment and a cellular compartment [1]. Experimental evidence
indicates that the concentration of substances in the space of Disse is very close to the
concentration of the substances within the sinusoids, since the fenestrate are large [22].
Lidocaine undergoes linear oxidative deethylation of the tertiary amine group and forms
monoethlglycinexylidide represented as MEGX. Lidocaine also undergoes hydroxylation
of the aromatic ring to form 3-hydroxylidocaine represented as 3-OHLID. These two are
the important metabolites of lidocaine.
2.2.1 Assumptions and Equations
(1) The general model consist of a series of repeating cellular-extracellular parallel
compartments.
(2) The sinusoids and the interstitial space of Disse are approximated as a single unit
called the extracellular compartment.
(3) The second compartment is the tissue or cellular compartment. This compartment is
assumed to behave as a homogenous pool for both reaction and efflux.
(4) Mass transfer takes place between these two compartment by kinetic rate constants
k12 and k21.
(5) Metabolites are formed in the cellular compartment.
(6) The number of composite compartment units does not influence the effectiveness of
this model in representing washout profiles of lidocaine and its metabolites. The reason
for this is that time to complete cellular uptake and release is significantly longer than the
time require to clear the extracellular network [I]. Therefore for our purpose we choose
N=1.
(7) Inputs and outputs are by way of the extracellular compartment only.
Figure 2.1 Compartment model for hepatic elimination of lidocaine and its metabolites (From B.A. Saville, M.R. Gray, and Y. K. Tam, "Experimental studies of transient mass transfer and reaction in the liver, Interpretation using a heteregenous compartmental model", J Pharm. Sci., 81(3), 265-271, 1992)
The model shown in Figure 2.1 to describe liver metabolism was first proposed by
Weisiger [22] and then modified by Saville [1]. There are 'N' number of extracellular and
cellular compartments arranged in series parallel order. Compartments '1' and '2'
represent the cellular and extracellular compartments respectively. The concentration of
10
cellular and extracellular compartments is represented as C1j and C2j respectively. 'Q' is
the blood flow in to and out of the cellular compartment. Cin is the input to the cellular
compartment. As seen in Figure 2.1 'j' is the number of compartments. For lidocaine
'j'=1. Mass transfer equations for Figure 2.1 are given as
V1j [ C1j / dt]= Q (Cin - C1j) - k12V1jC1j + k2Vl2jC2j (1)
V2j[dC2j/dt ] =kl2V1jClj - k2lV2jC2j - V2jkrC2j + V2jkfCp2j (2)
Where:
V1j = Volume of extracellular compartment (mL)
V2j = Volume of cellular compartment (mL)
Q = Volumetric flow rate through organ (L/min)
Cin = Input concentration (uM)
Clj = Concentration in extracellular compartment (p.M)
C2j = Concentration in cellular compartment (i.tM)
Cp2j = Precursor concentration in the cellular compartment (µM)
Cm2j = Metabolite concentration in the cellular compartment (p.M)
k12 = Rate constant of uptake in to liver cells from extracellular space (min-1)
k21 = Rate constant of release in to liver cells from extracellular space (min-1)
kf = Rate constant for intracellular formation (min-1)
kr = Rate constant for intracellular reaction (min-1)
Figure 2.2 Metabolites formation of lidocaine.
Figure 2.2 shows the reactions occurring in the cellular space when the species of interest
C2j reacts to form metabolites Cm2j. Metabolites are formed in the cellular compartment
by the kinetic rate constant kr. Depending on the rate constant, metabolite formation
occurs. If kr increases then metabolites form quickly and if kr decreases then metabolites
form slowly.
2.2.2 Calculations
Vlj and V2j are calculated as ;
Vlj = Vv / N; V2j = VC / N (3)
N is the number of compartmental units
N = 1 for Lidocaine and its metabolites
Vv is extracellular volume (rnL)
VC is Cellular volume (mL)
Vtot = VC + VV (4)
Where Vtot is the total physiologic volume of the liver (mL)
f he volume of the cellular region is nearly six times that of the extracellular region [1].
V2j = 6 * (V1j) (5)
Vtot = Wl/fl (6)
Where WI = Liver weight (g)
f l = Density of liver (g/L)
12
All experiments used rat livers isolated from male Sprague-Dawley rats [1]. Rats were
anesthetized with diethyl ether, and the liver was isolated by surgical cannulation of the
hepatic and portal veins and inferior vena cava. The bile duct was not cannulated. During
experiment, the organ was kept moist and at a physiological temperature. Concentrated
lidocaine was infused with a syringe pump. Samples of the inlet concentration were taken
every 10 min to verify the infusion rate and to ensure that a constant concentration at the
inlet was maintained. For washout experiments, lidocaine was infused until steady state
was reached. Drug infusion was then stopped. Samples of the liver effluent were
optioned every 6 seconds. Parameters for this model were estimated from the
experimental data [1, 22] at a standard set of conditions.
WI= 10 gms 11= 1 g/L
Therefore, Vtot = 10 mL (from equation (6))
Using equations (3), (4), and (5)
V1j 1.4285 mL and V2j = 8.571 mL
From [22] at standard test condition
Wl/Q = 0.33 but WI = 10 g Therefore Q = 30.30 L/min
Next kinetic rate constants were estimated. The most important consideration in
modeling the metabolism of lipophilic compounds in the liver is the proper estimation of
the kinetic time constant values. When the model was simulated on a 486 based personal
computer with VisSim (Visual solution Inc.,) [10] software with kinetic constant values
published in reference [1], it was found that the system was unstable. Kinetic rate
13
constants values given in reference [1] for k12 and 101 were 1206 = 830 min -1 and 46
39 min -1 respectively for lidocaine. When model was simulated with these kinetic rate
constant washout profile were unstable. Concentration was getting very high value in
range of thousands of micomolar. Also concentration was achieving negative values
which is impossible. With kinetic constant values published in reference [1] for lidocaine
metabolites MEGX and 3-OHLID concentration reached very high value and also
negative values.
2.2.3 Data Analysis
We then used the exponential peeling method [24] to obtain parameters which
characterized the experimental curves presented in reference [1]. We used these values in
the heterogeneous model.
Model equations (1) and (2) were solved analytically for N = 1. This solution was
compared with the exponential peeling equations and kinetic constants were estimated
which then allowed us to fit the experimental data. Table 2.1 gives the comparison of the
kinetic constant values which are given in the reference and values which were obtained
by the exponential peeling method. It is seen that there is vast difference in the values
between the two.
Table 2.1 Comparison of kinetic constants values for lidocaine given in the reference and predicted after exponential peeling method
Values given in the reference Predicted value
k12 = 1206 ± 83 min -I k12 = 9.7 min -1
Ic.7 ----- 46 ± 39 min -1 k21 = 0.48 min -
kr ----- 0.49 ± 0.16 min - kr = 0.49 min
14
By comparing this results, it was seen that the values for k12 and k21 were 100 times less
than those reported in reference [1]. Difference in the values of kinetic rate constant may
be due to typographical error. This was then repeated to obtain kinetic constant values for
the lidocaine metabolites. The rate constant values were changed accordingly and new
predicted values were obtained are shown in table 2.2.
Table 2.2 Predicted rate constant values for lidocaine metabolites
Rate constant MEGX 3-OHLID
.1(12 (min-I) 0.38 2.2
121 (min-1) 0.03 0.08
kr min- I) 0.24 0.35
Simulation was done using these parameters on a 486 based personal computer
using VisSim software [10]. It was observed that the system behaved in a predictable
way. These parameters values were then used in a sensitivity study and a range of values
were determined for lidocaine and each of its metabolites.
The results obtained were compared with the experimental results or data
presented in reference [1]. It was found that the washout profile was very similar to the
one given in the reference. Simulation curves are kept in the appendix A. It was found
that the heterogeneous model accounts for the slow elution of the drug and its
metabolites.
Encainide is a highly effective, well tolerated antiarrthymic agent [25]. We tried
to model metabolism of encainide in the liver but were unsuccessful due to reasons stated
below; (1) There are few models available for modeling encainide metabolism. (2) The
models that are available contain so many simplifying assumptions that they are not
15
useful for a heterogeneous organ such as liver [26,27]. Therefore, another antiarrthymic
drug procainamide was modeled [28-30]
2.3 Model for Procainamide
The model which was previously described for lidocaine was used to model
procainamide with appropriate changes. Figure 2.4 describes the model and it is
represented by differential equations (7) and (8). 'u' is the amount of procainamide
present in urine. This parameter is taken in to account as procainamide is excreted in
urine with sufficient quantity so that it must be considered. It should be noted that
metabolism and breakdown occurs in the extracellular compartment. Therefore rate
constant for breakdown is included in the extracellular concentration equation.
Breakdown of procainamide in the extracellular compartment is due to the enzymes
which inactivates it. These are the changes which differ from the lidocaine model.
V j [dC 1j / dt] = Q 2 V 21 V2j C2j-kr V1j Cij - u Cij(7)
V2j [dC2j /dt] = k12 V -101 V7
(8)
Values for the kinetic constants and other parameters used in equations (7) and (8) were
obtained from [3,28]. However, the same values of Q, V 1 j, V2j were used as in the
lidocaine model. The reason for this is that the extracellular and cellular volumes and
flow are expected to be similar for both drugs.
Figure 2.3 Metabolite Formation of Procainamide
Figure 2.4 Comparmiental model for hepatic elimination of procainamide
2.3.1 Calculations
Cm lj is the metabolite of
kr is the rate constant for metabolite formation
k12 = 9.35 min -1
k21 = 9.35 -min -1
kr = 0.0053 min -1
u = 0.0013 ml/min/mole
Cin = 20, 50, 70 mg/kg (single impulse injection)
u is the concentration of Procainamide in urine output
Cin is converted from mg/kg to
Molecular weight of Procainamide hydrochloride = 271.79
mg/kg--> 10 µg/kg (mol. wt.) --> µM
17
Using this formula different values of Cin are obtained;
5.4358 µM for 20 mg/kg
13.5895 µM for 50 mg/kg
19.0253 µM for 70 mg/kg
Using these data and equations (7) and (8) simulations of the pharmacokinetics of
the procainamide in the liver were done using VisSim software 110] on a 486 based
personal computer. A parameter sensitivity study was also done to determine the range of
values for the kinetic rate constants for which the model is stable. Results are placed in
chapter 3.
CHAPTER 3
RESULTS AND DISCUSSION
The following chapter discusses the results of the simulations. The range of kinetic
constants which can be used in the stable operating range and sensitivity study are
described in detail. The simulation results and block diagram of differential equations are
kept in appendix A.
3.1 Lidocaine and its Metabolites
3.1.1 Washout Simulations
The clearance of lidocaine and its metabolites MEGX and 3-OHLID from the liver were
simulated by loading the system with an initial concentration of each drug. For each case
an initial condition was set in the integration block [step( hepatic loading )] and no input
was given to the model to simulate washout experiments. It was observed that the
concentration decreased exponentially with time. These results matched the data [1] from
washout experiments after cessation of drug administration. Curve 1 shows these results.
The simulation profiles indicate that the concentration of lidocaine and its metabolites
declined in a first order fashion (Curve 2, 3, and 4). These results are consistent with the
results obtained by radiolabeled lidocaine and supports the conclusion that drug
transport and release in the liver are linear.
It is observed that lidocaine concentration is higher than its metabolites in both the
cellular and extracellular compartments because lidocaine is the major chemical (drug)
while MEGX and 3-OHLID are its metabolites. All three drugs take nearly the same
amount of time to "washout" of the liver. The model also predicts that MEGX and 3-
OHLID are eliminated faster than lidocaine.
18
19
33,2 Sensitivity Study
A parameter sensitivity study establishes the range of kinetic constants for which the
model is stable. It was determined that, if the sensitivity range is exceeded, the cellular
and/or extracellular concentrations become negative or approach zero. The sensitivity
study was done by using a steady state (step input) input along with an initial condition
(hepatic loading). This is similar to the experimental setup in which the perfused liver is
first infused with drug to reach a steady state condition and then the drug is administered
at a constant rate after some predetermined time period. The maximum values of the
kinetic constants were determined at the stage where the system response was different
from the normal behavior. A step input was given, the kinetic constant value of one
parameter was either increased or decreased and kinetic constant values of other
parameters were unchanged. Simulations were done and these steps were repeated until
abnormal behavior in the model was observed. When the cellular and extracellular
concentration decreased or diminished, instead of increasing, this define the maximum
value and sets the upper limit for the kinetic constants for the model. Similarly, the lower
limit is set when simulated extracellular and cellular concentrations approached zero or
became negligible. It was seen that the rate constant for uptake was significantly different
for each drug and was lowest for MEGX and highest for lidocaine. This shows that
lidocanie is the drug which is taken in from compartment 1 to 2 very fast compared with
its metabolites. Similarly, the rate constant for release of the drug from the cells differed
for each drug. It was lowest for MEGX and highest for lidocaine. This shows that MEGX
takes the longest time to come out of compartment 2 to 1 and lidocaine takes the shortest
time. Curve 5.A shows the washout profile of lidocaine and its metabolites taken from
reference [1]. Curve 5.B shows the simulation results of washout profiles for lidocaine
and its metabolites (no input). It is seen that Curve 5.A and Curve 5.B looks similar.
Curve 5.0 shows simulation results with a step input and with initial conditions. The
20
extracellular concentrations increased and attained a new steady state value in a short
period of time. Thereafter, the extracellular concentration remained constant. The
cellular concentration remained unchanged for a step input because input does not affect
cellular concentration directly.
3.1.2(a) Sensitivity Study of Lidocaine
Curves 6.A and Curve 6.13 (cellular concentration of lidocaine with less effective change)
show a simulation in which the liver is loaded with an initial concentration of lidocaine
and then a step input is applied at some predetermined time. It is observed that for
nominal values of the parameters the extracellular concentration first decreases
exponentially and then increases exponentially to attain a steady state value after a long
time. When the kinetic constants are set at their maximum values, it is seen that
extracellular concentration starts decreasing. Using minimum values of the kinetic
constants the extracellular concentration reaches a value nearly equal to the step input in
a very short period of time. When the kinetic constants are increased to their maximum
values, the cellular concentration decreases to a very low value (Curve 6.B). When the
minimum values of the kinetic constants were used there was no effect of the step input
on the cellular concentration. It remained constant for all time. Thus, no significant
changes were observed for cellular concentration with input. Hence, cellular
concentration curves are not included for lidocaine metabolites.
3.1.2(b) Sensitivity Study for MEGX and 3-OHLID
Curves 7, and 8 show the simulation for lidocaine metabolites MEGX and 3-OHLID
respectively. When the maximum value of the kinetic constants were used the
extracellular concentration of both metabolites starts decreasing, while cellular
concentration approached zero. At the lower limit for the kinetic constants the
extracellular concentration of both metabolites reached very high values equal to the
21
steady state input, whereas the cellular concentration of both metabolites remained
unchanged for the entire simulation time period. The reason for the unchanged cellular
concentration is that the kinetic constants are so low that they do not affect the cellular
concentration. In other words nothing comes in and nothing goes out of the cellular
compartment. The maximum values of the kinetic constant represent very fast rates of
uptake and release. This leads to concentrations in the extracellular compartment below
their initial value and negative values for the extracellular concentration. The ranges of
the kinetic constants are summarized in table 3.1. Table 3.1 shows the summary of values
of kr, k21, kr.
Table 3.1. The range of values for k12, k21, and kr for lidocaine and its metabolites
k17 min-1 k21 min-1 kr min-
Chemical Min Max Min Max Min Max
Lidocaine 0.0 18.1 0.0 8.75 0.0 11.75
MEGX 0.0 18.4 0.0 16.45 0.0 12.35
3-OHLID 0.0 18.58 0.0 13.0 0.0 2.75
Maximum values of k12 and k71 which can be used simultaneously are given in Table
3.2. kr value denotes the rate constant of breakdown which can be used along with these
maximum values to give a stable model response.
Table 3.2 Maximum values of both k12 and k71 which can be used together
Chemical k17 min t k21 min -1
kr min
Lidocaine 10.7 3.25 0.49
3-OHLID 10.7
7.76 0.35
MEGX 12.58 6.35 0.24
22
The volume of the cellular compartment is nearly six times that of the extracellular
compartment. Thus, at steady state there is close to thirty times more material in the
cellular space than in the extracellular space. This shows that cells can acts as a large
reservoir for the drug and its meatabolities. Hence, it takes time to fill the extracelllular
compartment (drug uptake). It takes time to empty the cellular compartment (drug
washout). Normal values are ones for which results of simulation are consistent with the
results published. Table 3.3 summarizes the normal values of kinetic rate constants.
Table 3.3 Normal value of kinetic constant for lidocaine and its metabolites
Chemical k12 min-1 k21 min-1 kr min-1
Lidocaine 9.7 0.48 0.49
MEGX 2.2 0.08 0.35
3-OHLID 0.38 0.03 0.24
It is seen that changing flow (Q) has no effect on extracellular or cellular
concentrations. The reason for this is that the transport process is barrier limited. There
are two types of transport processes (1) flow limited and (2) barrier limited.
Flow limited:
Transport out of the compartment into the extracellular space is very rapid.
Therefore if flow is increased more material is brought to the transport site and increases
the amount of material transported.
Barrier limited :
Transport out of the compartment into the extracellular space is very slow so it
does not change very much if more material is brought to the transport site by increasing
flow.
23
3.2 Procainamide Simulations
For the antiarrthymic drug procainamide the heterogeneous model used is similar to that
used for lidocaine. For procainamide, metabolism occurs in the cells and hence the
differential equations for the cellular and extracellular compartments are modified
accordingly. Procainamide elimination occurs in urine [30] and that factor is included in
the model. Urine output is added to the extracellular compartment. Procainamide
metabolites are not important so they were not considered. With these changes
differential equations (8) and (9) were simulated on a 486 based personal computer using
VisSim (Visual Solution Inc.) software. Procainamide simulations are divided into
washout profiles, simulations with input and sensitivity study of different kinetic rate
constant parameters. Each of them is described in detail.
3.2.1 Washout Profile
Washout of procainamide is simulated using an initial condition in the integration block
(similar to the lidocaine washout simulation). Curve 9 shows the simulation. As time
increases the extracellular and cellular concentrations of procainamide decrease
exponentially in a first order fashion. It is seen that drug elimination is faster in the
extracellular compartment than the cellular compartment, because more concentration
leaves the compartment than enters the compartment. The rate of cellular uptake and the
rate of cellular release are assumed to be same for procainamide because it is not lipid
soluble. Hence, the same amount of time is taken by cellular compartment to take up
procainamide and to release procainamide.
3.2.2 Simulation with Input
Curve 10 shows the simulation of a bolus injection of procainamide. This input is similar
to a pulse of very short duration. Both cellular and extracellular concentrations are shown
24
along with the input. In this simulation no procainamide is present initially in either of
the compartments. It is seen that the extracellular compartment acts as a reservoir and its
concentration increases rapidly with the impulse (bolus injection). The maximum value
attained by the extracellular compartment is nearly equal to the peak of the impulse and
thereafter, extracellular concentration decreases suddenly to a zero value. The rise in
concentration is exponential but the fall in concentration is not exponential. The reason
for this is that before the concentration reaches saturation level in the extracellular region
the impulse starts to decrease causing the extracellular concentration to decrease. The
cellular concentration acts exactly as any linear system should act. The cellular
concentration increases exponentially and reaches its steady state value during the time
period of the impulse. When the impulse starts decreasing the cellular concentration also
starts to decrease in an exponential way. The amount of procainamide in the cellular
compartment is smaller than that in the extracellular compartment, because the cellular
compartment volume is greater than the extracellular compartment volume. It is also seen
that the time taken to reach saturation level during the impulse is higher in the
extracellular compartment than in the cellular compartment.
3.2.3 Sensitivity Study for k12
Curves 11.A and 11.B show the results of the sensitivity study done on the liver model
for procainmide for k12. It shows the range rate of uptake from the extracellular
compartment to the cellular compartment for extracellular concentration and cellular
concentration respectively. Within the given range the Curve 11.A shows the simulations
of extracellular compartment for which model is stable. The minimum and maximum
curves represent the lower and upper limit of k12. It is seen that when k12 is reduced
significantly with respect to k21, the extracellular concentration reaches a value above
the impulse amplitude value. This is not a realistic occurrence, hence this sets the lower
limit for k12. During the decreasing portion of the impulse it is seen that the extracellular
25
concentration decreases, then increases and then decays exponentially. This is similar to
an overshoot in an electrical system. On the other hand when k12 is increased and made
much larger than k21, it is seen that the extracellular concentration reaches negative
values when the impulse starts decreasing. This is not physically possible so this sets the
upper limit for k12. Curve 10.B shows the curve for cellular concentration with different
values of k12 with respect to k21. It is observed that for lower value of k12 cellular
concentrate becomes very low, while for higher values of k12, cellular concentration
reaches very high values.
3.2.4 Sensitivity Study for k21
Curves 12.A and 12.B show the effects of changes in the rate constants k21 with an
impulse as input, with no initial condition on cellular and extracellular concentrations
respectively. Curve 12.A shows the simulation results when the kinetic constant k21 is
changed for the extracellular compartment (k21 is decreased compare to k12). This is
physically represented as a longer time for the drug to be eliminated from the cellular
compartment. Simulating this condition, it was found that during the increasing part of
the impulse the cellular concentration increased in a linear manner and during the
decreasing phase of the impulse the cellular concentration decreased exponentially. It
was seen that a longer time was taken before the whole drug was completely eliminated
from the cellular compartment (Curve 12.B). This is consistent with the physical
interpretation. With k21 increased it is seen that during the increasing phase of the
impulse period, the cellular concentration reaches its steady state value quickly and then
the cellular concentration remains constant. During the decreasing phase of the impulse
the cellular concentration decreases rapidly. When the nominal value of k71 was used,
cellular concentration increases exponentially, attains a steady state value at the end of
the impulse period and then decreases exponentially with the decreasing period of the
impulse. This is similar to any stable linear system response.
26
Curve 12.A shows the effect of k21 for different ranges of extracellular
concentration. At the maximum value of rate constant k21, the extracellular
concentration increases to a higher value than the input, while for the minimum value of
k21 the extracellular concentration attains negative values. For the nominal value of k21,
the extracellular concentration increases during the increasing phase of the impulse and
then decreases during the decreasing phase of the impulse.
3.2.5 Sensitivity Study for 'u'
Curve 13 shows the results of the simulations for the range of the kinetic constant 'u'
(urine). For increasing values of 'u' the extracellular concentration increases
exponentially and before it reaches its steady state value, the impulse starts decreasing.
The extracellular concentration then drops and returns to zero. At this value of 'u' the
extracellular concentration attains higher value than the input. We consider this to define
the maximum value of 'u' for this model. The curve for the nominal value of 'u' shows the
action of a stable linear system. There is no lower limit of 'u' for extracelluar
concentration. Cellular concentration changes less than 5% with changes in the value of
'u', since 'u' does not effect cellular concentration directly.
3.2.6 Sensitivity Study for kr
The results of the sensitivity study of the rate constant of breakdown of the drug (kr) are
shown in Curve 14. It is seen that when kr is made very high the extracellular
concentration attains negative values for the decreasing phase of the impulse. This is
overshoot and it is the characteristic of an underdamped system. Concentration can never
attain negative values, thus this sets the upper limit for Kr. Physically increasing kr
means that the breakdown of the drug is taking place faster than it is being formed. There
is no lower limit for kr because, if kr is made very small a longer time is taken for its
27
metabolites to form and hence this will delay the formation of other metabolites. Cellular
concentration is less affected by changes in the rate constant of breakdown kr.
Thus, we have range of kinetic constants for procainamide which can be used for
stable simulation of the model. Also there are normal limits during which the model
behavior was similar to the predicted one. Table 3.4 summarizes the range of values for
parameters for procainamide.
Table 3.4 Procainamide simulations results with different ranges.
Minimum Normal Maximum
ki2 min-1 2.95 9.35 14.25
k21 min-1 0.5 9 35 20.5
u ml/min/mole 0 0.0013 6.78
kr min-1 11 0 I
0.0053 4.65
CHAPTER 4
CONCLUSIONS AND SUGGESTIONS
4.1 For Lidocaine and its Metabolites
The heterogeneous model represented by equations (1) and (2) is appropriate to
describe the metabolites of the lipophilic compound lidocaine and its metabolites.
Simulations were done for lidocaine and its metabolites and results obtained were
consistent with published data [1]. Washout profiles of lidocaine and its metabolites
exhibited a first order decline. With a step input the model behaved in a linear manner,
for lidocaine and its metabolites. The kinetic constant for cellular uptake of lidocaine was
higher than the kinetic constant for extracellular uptake. This states that lidocaine and its
metabolites are distributed faster in the cellular region than in the blood compartment
(since lidocaine is lipid soluble). The rate of conversion of lidocaine within the cells is
much slower than the rate of release from the cells. Thus, the amount of lidocanie
converted to metabolites will be significantly less than the amount of lidocaine
transported from the cellular compartment.
4.2 For Procainamide
For the antiarrthymic drug procainamide, it was found that the rate constant of uptake
from the extracellular compartment to the cellular compartment and the rate of release
were identical (procainamide is not lipid soluble). Sensitivity study estimated the upper
and lower limits for each rate constants. The simulations done provided insight of kinetic
constants and metabolism of procainamide in liver.
28
29
4.3 Future Work
The heterogeneous model presented in this thesis can be readily modified to include
enzyme inactivation. Also the model can be used to study the interactions of two or more
drugs if the interaction constants can be obtained. It will be interesting for the model to
be tested for simulating concentrations of toxic chemicals such as halogenated
hydrocarbons as well as other critical drugs. The heterogeneous model developed for
local anesthetic drug lidocaine can be used along with other models developed for
cadiovascular, respiratory and cerebral systems to simulate the overall effect of lidocaine
on the body. In this manner other drugs and toxic chemicals such as halogenated
hydrocarbon metabolism can be modeled and their effects can be simulated. The
heterogeneous model is capable of describing the time dependent profile of a variety of
compounds.
APPENDIX A
SIMULATION CURVES
Curve Page
Figure Al Block Diagram of Extracellular Compartment of Lidocaine 32
Figure A2 Block Diagram of Cellular Compaitment of Lidocaine 33
Figure A3 Block Diagram of Extracellular Compartment of Procainamide 34
Figure A4 Block Diagram of Cellular Compartment of Procainamide 35
Curve 1 Washout Profile of Lidocaine from reference [1] 36
Curve 2 Simulation of Washout Profile of Lidocaine 37
Curve 3 Simulation of Washout Profile of MEGX 38
Curve 4 Simulation of Washout Profile of 3-OHLID 39
Curve 5.A Washout Profile of Liodcaine and its Metabolities from reference [1] . . 40
Curve 5.B Simulation of Washout Profile of Liodcaine and its Metabolities 41
Curve 5.0 Simulation of Washout Profile of Liodcaine and its Metabolities with Input 42
Curve 6.A Sensitivity Study of Lidocaine for Extracellular Compartment 43
Curve 6.B Sensitivity Study of Lidocaine for Cellular Compartment 44
Curve 7 Sensitivity Study of MEGX 45
Curve 8 Sensitivity Study of 3-OHLID 46
Curve 9 Washout Profile for Procainamide 47
Curve 10 Simulation of Bolus Injection to Extracellular Compartment for Procainamide 48
Curve 11.A Sensitivity Study of k12 for Procainamide for
Extracellular Concentration. 49
30
SIMULATION CURVES (Continued)
Curve Page
Curve 11.B Sensitivity Study of k12 for Procainamide for Cellular Concentration . 50
Curve 12.A Sensitivity Study of k21 for Procainamide for Cellular Concentration . .51
Curve 12.B Sensitivity Study of k21 for Procainamide for
Extracellular Concentration 52
Curve 13 Sensitivity Study of 'u' 53
Curve 14 Sensitivity Study of kr 54
Figure Al Block Diagram of Extracellular Compartment of Lidocaine
Figure A2 Block Diagram of Cellular Compartment of Lidocaine
Figure A3 Block Diagram of Extracellular Compartment of Procainamide
Figure A4 Block Diagram of Cellular Compartment of Procainamide
36
Curve 1 Washout Profile of MEGX from reference [1]
37
Cellular Cnncentration
Curve 2 Simulation of Washout Profile of Lidocaine
38
Curve 3 Simulation of Washout Profile of MEGX .
39
Curve 4 Simulation of Washout Profile of 3-OHLID
40
Curve 5.A Washout Profile of Liodcaine and its Metabolities from reference [1]
41
Curve 5.B Simulation of Washout Profile of Liodcaine and its Metabolities
42
Curve 5.0 Simulation of Washout Pronie of Liodcaine and its Metabolities with Input
43
Curve 6.A Sensitivity Study of Lidocaine for Extracellular Compartment
44
Curve 6.B Sensitivity Study of Lidocaine for Cellular Compartment
45
Curve 7 Sensitivity Study of MEGX
46
Curve 8 Sensitivity Study of 3-OHLID
Curve 9 Washout Profile for Procainamide
48
Curve 10 Simulation of Bolus Tnjection to Extracellular Compartment for Procainamide
49
Curve 11.A Sensitivity Study of k12 for Procainamide for Extracellular Concentration
50
Curve 11.B Sensitivity Study of k12 for Procainamide for Cellular Concentration
51
Carve 12.A Sensitivity Study of k21 for Procainamide for Cellular Concentration
52
Curve 12.B Sensitivity Study of k21 for Procainamide for Extracellular Concentration
53
Curve 13 Sensitivity Study of 'u'
54
Curve 14 Sensitivity Study of kr.
APPENDIX B
VISSIM DESCRIPTION
VisSim (Visual Solution Inc., Westford, M.A.) is a powerful set of software with the
ability to design, simulate, plot, and debug within a single interactive environment.
Models and systems are built using blocks. Blocks are connected to each other and to
displays using flex wires. Blocks and flex wires are the primary tools of VisSim.
There are two types of blocks (1) standard blocks and (2) compound blocks.
Standard blocks includes (l) Addition block, (2) Arithmetic block, (3) Integration, (4)
Non-linear, (5) Boolean etc. A compound block is the combination of more than one
standard block. There are also signal producing and signal consuming blocks. The signal
consuming block include plot which is the output of any system in the form of a graph
with time, frequency or parameter varying on 'y' axis. Signal producer block includes
step, sinusoids, unknown etc. Any of these blocks can be used to represent a physical
situation or to produce a signal. An example of this is that a bolus injection of a drug can
be mathematically represented as an impulse. Consider the case, where a bolus injection
is given between time 3 to 5 sec. For any other time the signal produced is zero. This is
represented mathematically as shown below;
y = I for 3 < t < 5
else y = 0
This condition can be simulated using two step blocks and a summer block. By setting
the proper time for the step block and adding one block (with t=3) and subtracting one
(with t=5) a pulse of fixed duration of 2 second is produced. A step input can be used
whenever the input is constant to the system after some predetermined time period. There
are output connector tabs with triangular shapes which helps the user to view the
direction of the signal. Each block has two
55
(Block Diagram of A(d2x / dt2) - B(dx/dt) + Cx = 0)
57
types of signals (I) input signal or coming in signal and (2) output signal or exiting out
signal.
In modeling the hepatic system for drug elimination, there are two differential
equations; one for the cellular compartment and one for the extracellular compartment.
These equations were represented by blocks on the VisSim (Visual Simulation Inc.,
Westford, M.A.) software. VisSim has the ability to solve several types of differential
equations, including ordinary differential equations (OBE's). VisSim solves ODE's by
using an integration operator. The example below shows how VisSim solves a second
order differential equation. Let the equation be;
A(d2x /dt 2 ) - B(dx/dt) + Cx = 0
Since integration is more numerically stable than differentiation, the above equation can
be represented in terms of integrators. If we use 1/S (Laplace Operator) as a short hand
notation for an integrator, then x(t) can be represented by;
x(t) = I /S [dx/dt] = (1/S) [1/S (d 2x) /(dt2)]
The original equation can be written in modified form as;
2x / d
/ A [ B (dx/dt) - Cx
Figure on the opposite page shows block diagram of this differential equation. This
equation is implemented on VisSim by wiring the output of the x and dx/dt variable
blocks through two gain blocks (which represent 'C' and 'B' respectively). These two are
then given as input to the summing block with a negative sign on the signal coming from
the 'x' variable and a positive sign on the signal coming from the 'dx' variable. The output
58
of the summing block is divided by the constant block whose value is equal to 'A'. The
output of this block is d2x/dt2. This diagram represents a closed loop system. From this
loop values for x(t), dx/dt, d2x/dt2 can be tapped off and displayed using the signal
consumer plot. Different values can be entered in appropriate blocks to simulate different
sets of initial condition and system parameters.
The procedure to built a file and simulate a differential equation is as follows;
(1) From file menu create a new file
(2) From blocks in the menu create appropriate blocks, then connect them properly using
flex wires so that they represent the differential equation.
( ) Insert proper values for parameters and initial conditions in the appropriate blocks.
(4) Check that the system is closed loop and all signs are properly matched.
(5) Create plot from the block menu and connect the outputs of the differential equations
to the plot.
(6) Go to edit menu and simulate the differential equation.
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